(Requires mathematical statistics.) Maximum likelihood estimation of N in large samples.
Suppose that n1 of the N fish in a lake are marked. An SRS of n2 fish is then taken, and m of those fish are found to be marked. Assume that N, n1, and n2 are all “large.” Then the probability that m of the fish in the sample are marked is approximately:

a. Show that Ṅ= n1n2 / m is the maximum likelihood estimator of N.
b. Using maximum likelihood theory, show that the asymptotic variance of Ṅ is approximately N2 (N − n1) / (n1n2).